Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$

TL;DR

This paper proves Lusztig's conjectures for the affine Weyl group of type C_2 for all positive weights, using balanced representations and an asymptotic Plancherel theorem, completing the conjectures for low-rank affine groups.

Contribution

It introduces the notion of balanced systems of cell representations and establishes an asymptotic Plancherel theorem for C_2, leading to a full proof of Lusztig's conjectures in this case.

Findings

Proved Lusztig's conjectures P1-P15 for C_2 with all positive weights.

Developed the concept of balanced systems of cell representations.

Established an asymptotic Plancherel theorem for type C_2.

Abstract

We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl group of type $\tilde{C}_2$ for all choices of positive weight function. Our approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of a `balanced system of cell representations'. Once this system is established roughly half of the conjectures ${\bf P1}$-${\bf P15}$ follow. Next we establish an `asymptotic Plancherel Theorem' for type $\tilde{C}_2$, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank $1$ and $2$ affine Weyl groups for all choices of parameters.